So what happens now, with the primary season ending, and the Tea Party having defined it? Does the Tea Party remake the G.O.P. in its image, staging a “hostile takeover,” as Matt Kibbe, the president of FreedomWorks, the libertarian advocacy group, urged activists rallying outside the Capitol last weekend to do? Or will the Republican Party co-opt the Tea Party, as Trent Lott, a former leader of the Senate Republicans, said it must?

The embodiment of this question might be Senator Jim DeMint, the South Carolina Republican who has made himself and his Senate Conservatives Fund a kind of Tea Party Good Housekeeping seal of approval. Sitting at the intersection of the Republican Party and the Tea Party, Mr. DeMint could be a model for how the two might co-exist — or an example of how the drive for ideological purity could turn the Republicans into a niche party.

Mr. DeMint insists that he would rather have a pure conservative minority than a majority full of Republicans in name only. Republicans say Ms. O’Donnell will not win in the general election; Mr. DeMint says her opponent, Representative Michael N. Castle, would have just been another vote for President Obama and the Democrats’ agenda.

“The people of this country are uniting around these core principles of limited government,” he said, “and if the party can reflect that, we’ll be the big-tent majority party that we have been talking about for years. And anyone who says that we would be in the majority if we fielded a bunch of moderate candidates, just doesn’t understand what’s happening in America.”

The Republican tasked with trying to win enough seats to gain the party the majority in the Senate, John Cornyn of Texas, sees it differently.

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Senator Jim DeMint has embraced the ideological purity that characterizes many candidates with Tea Party backing.Credit
Stephen Crowley/The New York Times

“Jim has made it clear he wants to support the most conservative candidates,” Senator Cornyn said. “I believe we should find the most conservative candidates who can get elected. That’s a nuance, but it’s important.”

Even some of the primaries that Tea Party candidates lost suggest how much the Tea Party sentiment has already pushed Republicans to the right.

In Tuesday’s Republican primary in New Hampshire, for example, two Tea Party candidates in the Second Congressional District lost to Charlie Bass, a former congressman swept out in the Democratic wave of 2006. Mr. Bass was once known as the classic New England moderate. But to win the nomination this year, he campaigned far to the right — so far that The Concord Monitor editorialized, “It will take such a long way back to the middle that he’d better pack a lunch.”

Democrats are certainly counting on the Republicans’ taking a very long trip to a very remote region of the right.

Facing dispiriting polls and an enthusiasm gap that favors the Republicans, they rejoiced at the victories of Tea Party insurgents over establishment candidates Tuesday.

“When Mike Castle loses in Delaware, that’s a sign that moderates are no longer welcome,” said Representative Chris Van Hollen, the chairman of the Democratic Congressional Campaign Committee. “They’ve now become the narrowest of ideological parties, and I do think that’s going to alarm centrist moderate voters. It’s those centrist moderate voters that determine the outcome in these swing districts.”

“The Republican Party had hoped to harness the energy of the Tea Party movement,” Mr. Van Hollen said. “Now they have a runaway horse that they can’t control.”

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The Republican leaders Mitch McConnell, center, and John Boehner, right (with Harry Reid, the Senate majority leader).Credit
Drew Angerer/The New York Times

But Mr. Cornyn, who has been on the receiving end of anti-establishment anger, argued that the Tea Party had helped Republicans in one important respect, by moving the debate away from social issues. While Tea Party supporters tend to be socially conservative on issues like same-sex marriage and abortion, most say they don’t want to talk about them; they believe that by spending so much time on those issues, the Republican Party failed to focus on fiscal conservatism.

While social issues tend to be polarizing, Republicans can win on economic issues, Mr. Cornyn said, because the Democrats have been in charge as the economy has gone south.

“As I’ve traveled,” he said, “I’ve talked to a lot of folks who are basically independents who say: I’m fine with the Republicans as long as we’re talking about fiscal responsibility. Where I go off the reservation is when you talk about social issues.”

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Still, if the Republicans win a majority in the House or Senate, the pressure will increase to develop coherent policy.

“Having to produce a work product in the majority is much more difficult than saying no when you’re in the minority,” said Tom Davis, a former Republican congressman from Virginia.

“I don’t even think the Tea Party agrees with itself,” said John Feehery, a Republican consultant and former top aide to House Republicans. “Our message right now is not an aspirational message.”

But Mr. Feehery argued that the debate would not be settled on Capitol Hill. There will be a year of excitement about Congress, he said, until the party begins focusing on picking its presidential candidates.

“That,” he said, “will really litigate all the internal inconsistencies within the party.”

A version of this news analysis appears in print on September 16, 2010, on Page A22 of the New York edition with the headline: G.O.P. Gets a Partner, But Who Will Lead?. Order Reprints|Today's Paper|Subscribe